Constant tension control system for a dc reel drive motor

ABSTRACT

A constant tension system for a direct current reel drive motor operating to wind or unwind a coil of strip material moving at a line speed. A coil diameter signal D is derived which is a function of the instantaneous coil diameter, the coil having a finite diameter Do when the reel drive motor is at the weakest motor field setting. A motor field controller coupled to receive the coil diameter signal regulates the field flux  phi  so that when D&gt;Do,  phi  is a function of D, and when D&lt;/=Do,  phi  is a constant. An armature current controller is coupled to receive the coil diameter signal D and to control the armature current Ia so that when D&gt;Do, Ia is constant, and when D&lt;/=Do to enable the operator to reduce line speed so that the reel drive motor will not exceed its highest rated voltage and speed.

United States Patent 1 Pittner CONSTANT TENSION CONTROL SYSTEM FOR A DCREEL DRIVE MOTOR TENSION ROLLS [75] Inventor: John Richard Pittner,Pittsburgh, Pa.

' [73] Assignee: Westinghouse Electric Corporation,

Pittsburgh, Pa.

[22] Filed: Jan. 3, 1972 [21] Appl. No.: 214,808

[51] Int. Cl B59h 59/38 [58] Field of Search 318/6, 7; 242/7544 [56]References Cited UNITED STATES PATENTS 3,223,906 12/1965 Dinger 318/73,192,482 6/1965 Long 318/6 UX 2,021,757 11/1935 Umansky 318/7 X2,366,148 12/1944 Montgomery 318/7 X 3,348,107 10/1967 Hamby 318/63,280,393 10/1966 Crossley 318/6 3,018,978 1/1962 Graneau et a1. 318/6 XQ) STRIP TENSION F MANDREL 3? g g I GEARING '6 K TM 18 D.C. REEL MOTOR TTACHOMETER GENERATOR Primary Examiner-T. E. Lynch Attorney-F. H.Henson.J.J. Wood et a1.

[57] ABSTRACT A constant tension system for a direct current reel drivemotor operating to wind or unwind a coil of strip material moving at aline speed. A coil diameter signal D is derived which is a function ofthe instantaneous coil diameter, the coil having a finite diameter Dowhen the reel drive motor is at the weakest motor field setting. A motorfield controller coupled to receive the coil diameter signal regulatesthe field flux 4) so that when D Do, d) is a function of D, and when DDo, is a constant. An armature current controller is coupled to receivethe coil diameter signal D and to control the armature current Ia sothat when D D0, la is constant, and when D 5 Do to enable the operatorto reduce line speed so that the reel drive motor will not ex-v ceed itshighest rated voltage and speed.

5 Claims, 5 Drawing Figures 22 OPERATO D CONSOLE PAIENIEDJIII alum,749,9 8

SHEU 2 0F 2 THYRISTOR GATING AMPLIFIER EQ'L 'E FUNCTION POWER GENERATORSOURCE |8 (F.G.)) B4 82 CURRENT If E SENSOR FIG.3

7.. D. '36- 9 t- E4- D 03- (9 m F.G. INPUT vou's FIG.4

,124 CURRENT SENSOR GATING AMPLIFIER FlG.5

BACKGROUND OF THE INVENTION l. Field of the Invention? The inventionrelates to a constant tension control system for a direct current a'reeldrive motor operating to wind or unwind a coilo strip material moving ata preset line speed 2. Description of thePrio'r Art The prior arthaszprovided a variety of techniques for controlling a direct currentreel drive motor so as to ensure constant tension in a coil of stripmaterial thatis being wound or unwound. For the most part thesetechniques have worked well b iit the problem remains that they arecostly. The present invention provides a system which may beappliedhomost process lines to enable an increase in the reel diameter rangebeyond the direct current motor speed range which is available by fieldweakening, while stil maintaining constant tension. The horsepowerrequirements for the reel drive motor do not have'to be increased, ifthe line speed is lowered slightly below optimum at the smaller reeldiameters. This reduction in line speed at the smaller coil diameters ismodest and is tolerable for most process line applications.

SUMMARY OF THE INVENTION A constant tension control system isprovided'for'a diameter signal means for receiving the'diameter signal Dfor controlling the field flux of said reel drive motor so that the fluxis a function of the instantaneous coil diameter'D when D Do and theflux is constant when D "D0. Means are also coupled to the diametersignal .means for receiving the diameter signal D and for controllingthe armature current la of the reel drive motor so that [a is a constantwhen D. D and is a function of D when D D0. Means are also coupled tothe diameter signal means to provide a signal to enable the line speedto be reduced when D Do.

BRIEF DESCRIPTION OF TH DRAWINGS v FIG. 1 is a schematic illustratingone environmental setting for the constant tension control system of theinstant invention; Y 1

FIG. 2 is an electrical schematic showing the diameter memory anddiameter signal circuitry utilized in constant tension control system ofthe invention;

FIG. 3 is an electrical schematic showing the motor field flux controlcircuitry utilized in'the-constant tension control system of theinvention; H

FIG. 4 is a diagram depicting output voltage vs input voltage for thefunction generator used in the circuitry of FIG. 3; and

' GENERAL CONSIDERATIONS Referring'now to FIG. 1, a strip of materialpassing through tension rolls as being wound on a mandrel 12 to form acoil 14. As noted in FIG. 1, the coil has a variable diameter D.Themandrel 12 is coupled, through gearing shown symbolically at 16, to adc. reel drive motor 18. A tachometer genrator 20 is connected to theshaft of motor 18. The output signal of the ta- .chometer generator isutilized at several points throughout the system, but here in FIG. I itis shown being coupled to the; operator'saconsole 22 to enable the reeldrive motor speed to be monitored. The Operators console 22 is connectedto enable the speed of the line speed pacer motor (unnumbered) to becontrolled. (The line speed pacer motor sets the speed for the entireprocess line. Reel drive motor 18 essentially acts as a drag machineduring unwind, but as a motoring machine during wind-up) The torque ofthe reel and the torque of the motor are indicated at T, and Trespectively.

In order to understand the rationale upon which the invention rests, thefollowing mathematical analysis will be presented.

F T /r where F strip tension (force) Tr reel torque reel radius But bydefinition r D12 where D reel diameter Substituting for 'r in (l) T /D i7 (3') T FD/2 Neglecting losses, under steady state. conditions, thereel torque T transmitted "through the gearing is balanced by the motortorque T Thus K, mo'tor'torque constant the field flux Ia thearmaturecurrent Substituting the relationship of equations (4) and (5)in equation (6).

Solving for strip tension F where K, is a constant of proportionalitySubstituting the latter equation in equation (8),

Thus the strip tension F depends only on the operator setting for thearmature current reference la. In contemplation of this invention,constant tension will be obtained by flux control between full field andweak field.

At weak field no further weaking of the field is permissible. At thecoil diameter Do at weak field, the instant invention switches tocontrol of the armature current as a functionof the coil diameter, tomaintain constant tension, the flux being held constant. Themathematical considerations are as follows:

la K D for D D where D the instantaneous diameter D0 the coil diametercorresponding to the weakest motor field fiux. K a constant ofproportionality which includes the I operators reference settingSubstituting (11) in equation (8) F= (2 K/Kl) X (dz K D/D) At weak fieldflux, d) itself is kept constant, so that the tension F depends only onthe setting of In this mode of operation, i.e., 1a is varied as afunction of D, the line speed may have to be adjusted. This is broughtabout by the fact that in a process line there is a motor (line placesetter motor) drive which sets the speed of the line this is inaccordance with the line speed setting which the operator selects at theoperator console 22. The current regualtor reel motor 18 drags or setsup a counter force in the material which establishes the tension. Whenthe weak field is reached, the reel motor is then runningat the highestspeed. At this point the la is adjusted downward if in thepay-out modeor upward if in the wind-up mode. The reel drive motor operates atwhatever voltage is necessary to maintain the line speed. Thus at weakfield unless the reel motor 18 I and its power supply can temporarilyoperate at an over-voltage condition, the line speed must be reducedslightly.

DESCRIPTION OF THE PREFERRED EMBODIMENT The system of the inventionmaintains constant tension control in a moving strip of material in thepayout as well as winding-up modes by means of (a) field weakeningregulation (circuitry of FIG. 3) and (b) armature current regulation(circuitry of FIG. 5). As has been indicated under the caption GeneralConsiderations, the strip tension F is a function of the reel diameterD, and more specifically it varies inversely as the diameter D. In orderthen to maintain constant tension, one must know D at all times.Moreover, since there are occasions when the process line must betemporarily stopped, it is imperative thatthe diameter D at the momentof stopping be remembered." The circuitry of FIG. 2 will accomplishthese dual objectives.

Referring now to FIG. 2, the, reel speed signal (R.P.M.) obtained fromthe tachometer generator 20 is applied as'a divisor input to a divider24. The line speed signal (which is a function of the line speed in feetper minute (FPM) is fed as a dividend input to the divider 24. Thequotient output 26 from the divider 24 isapplied to an operationalamplifier, indicated generally at 28, through a resistor 30. Informationconcerning the start-up diameter of the coil 14 is applied to theoperational amplifier 28 by meansof a potentiometer- 32 connectedbetween a positive source of dc. voltage (+24v) and ground, the wiperbeing serially connected through normally open contacts lCRl, andresistor 34 to the amplifier input.

The relays described in this application are identified as follows. Thecoils are numbered lCR, 2CR and 3CR.

The contact pairs associated with the respective coils.

are identified for example as-3CR1, 3CR2, 3CR3, the 3 preceeding the CRidentifying the third contact relay, the numeral following CRidentifying the particular contact pair viz first (1), second (2), third(3); the bar beneath the identifying numeration indicates that thecontact is nonnally closed, while no bar signifies that the contacts arenormally open.

The operational amplifier 28 includes a high gain amplifier 36 having afeedback path contianing contacts 3CR1 and 1CR2, in parallel with animpedance 38. The output of the amplifier 36 is connected to a motoroperated potentiometer indicated generally at 40, having potentiometers44, 46, the wiper contacts of which are arranged to the rotationallydisplaced in unison in accordance with the rotation of the motor 42. Thewiper of potentiometer 44 is connected to the. input of the amplifier 36 through resistor48. The wiper'ofpotentiometer 46 is connected throughthe contacts 3CR2 to the output 50. The wiper of potentiometer 46 may beused to provide a signal for the operator to monitor when D0 is reachedor a separate potentiometer may be mounted on the common shaft ofthe-motor operated potentiometer 40. Completing the description of'thediameter memory and signal circuitry, the relay ICR is energized throughcontacts 52 which are momentarily closed on signal from the logiccircuitry 54.

The diameter signal D of FIG. 2 is applied simultaneously to thecircuitry of both FIGS. 3 and 5. First considering FIG. 3, the diametersignal is applied to diode 56 poled as shown, through resistor 58 to anoperational amplifier indicated generally at 60. A diode 62 has itscathode connected to node A, while its anode is connected to the wipercontact of a potentiometer and ground. I

The operational amplifier 60 is a proportionalintegralcontroller-fcomprising a high gain amplifier having a feedbacklpathwhich includes serially connected capacitor 6'6 and iesistor 68 Thegain of the amplifier is adjustable by means of the series connectedpotentiometer-resistor combination indicated generally at 70. The outputsignal from the operational amplifier 60 is applied to agatingramplifier 72, the output of which is connected to a -thyristorpower module 74. The power module74 is coupled to a power source 76which may be single phase or three phase a.c. depending upon powerrequirements for the system. The output of the T.P.M. 74 is applied tothe shunt field 78 of the reel motor 18. The shunt field 78 is arrangedin series with a current sensing resistor 80. A current sensor signal 1,derived from resistor 80 is applied to a current sensor 82 whichdevelops a voltage which is a function of the field current. The outputof the current sensor 82 is applied to a function generator 84 whichdevelops a signal which is a function of flux d), which signal is thenapplied to the motor flux controller 60 through a resistor 86.

The diameter signal D is also fed to the FIG. circuit through a diode 90having its anode connected to a node identified at B. A potentiometer 92is connected between a positive source of d.c. (+24v) and ground, thewiper of which is connected to the cathode'of a diode 94, whose anode isconnected to node B. A resistor 96 is connected at one end we source ofd.c. potential (+24v) the other end being connected to node B. Anoperators tension potentiometer 98 is connected between node B andground. The wiper of potentiometer 98 is connected to an'operationalamplifier indicated generallyat 100, through a resistor 102. An inertialcompensation signal Ic is applied to therproportional integralcontroller100 through a resistor 88. The operational amplifier 100, arranged tooperate as a proportional integral controller, comprises-an amplifier104 having an output resitor-potentiometer 106,108 connected between theoutput and ground. The wiper of potentiometer 108 is-connected in afeedback path pacitor 112.

The output of the proportional integral controller 100 is applied to agating amplifier 114 which is coupled to a thyristor power module(T.P.M.)indicated symbolically at 116. The thyristorrpower module 116 isconnected to a power source 118 which'may be single phase or three phasea.c. The output of the thyristor power module 1 16 is applied to thearmature 120 of the reel drive motor 18. In series with the-armature 120is a current sensingresistor 122. The armature current through resistor122 is applied to current sensor 124 which developes a voltage signa'lwhich is fed back to the operational amplifier 100 through resistor 126.A relay 2CR is connected in shunt with-the armature 120. The relay 3CRis connected to a relay power source +V through normally open contacts2CR1.

OPERATION OF THE CONSTANT TENSION CONTROL SYSTEM ment by adjusting thepotentiometer 98 (FIG. 5). The diameter. size of the reel14 (FIG. l)-tobe payed out is knownto the operator and this information requires acertain definite setting for the potentiometer 32 (FIG. 2). In the steelmill, the reel 14 (FIG. 1) is first mounted on a mandrel '16 which issegmented and adapted to collapse so as to fit inside various inner reeldiameters. When the mandrel 12 is released it presses outward forcingitself tightly against the inner diameter of the reel. By means ofsensors (not shown) the logic circuitry 54 (FIG. 2) then sends a signalmomentarily closing contacts 52 thereby connecting power to relay lCR.Upon the energization of relay lCR: (a) contacts pair lCRl (FIG. 2)closes and the voltage corresponding to the operator setting forpotentiometer 32 is applied as an input to operational amplifier 28, and(b) contact pair 1CR2 opens removing the short in the amplifier toenable the actuation of the motor 42. The come-to-rest setting of thepotentiometers 44,46 represents an analog voltage representative of theinitial diameter of the reel 14. g

At low speeds of the reel motor 18, the quotient output-26 of thedivider 24 is unreliable, (accordingly the divider is not utilized untilthe reel motor 18 reaches sufficient speed to make the quotient output26 reliable). In the meanwhile the output signal D is that of theinitial diameter. As the reel motor 18 increases in speed, a point isreached where the quotient output 26 of the divider 24-is reliable; thispoint maybe identified in that the armature (FIGLS) develops a voltageof a sufficient magnitude to energize relay ZCR. The sole contact of 2CRviz 2CR1 closes applying power to relay 3CR. The energization of relay3CR performs three functions: (a) 3CR1 opens removing the shorton theoperational amplifier 28, and the quotient signal26 is applied to theoperational amplifier 2 8 to cause the motor operated potentiometer 40to be actuated; (0) contacts '3CR2 open isolating the motor operated p0-tentiometer 40 from the circuits of FIGS. 3 and 5; and (c) contact pair3CR3 closes so that the quotient signal 26 is fed directly to theoutput. Thus the setting of the wiper 46 serves to remember" thediameter D should the process line be stopped for any reason. Also itshould be observed that the quotient signal 26 is fed directly to thecircuitry of FIGS. 3 and 5 through now closed contacts 3CR3. I i i Abrief consideration of the nature of the quotient signal 26 is in order,

mm. number of feet ['CVOIUIlOllS l revolution min.

One revolution the instantaneous circumference of the reel. Since thecircumference 1r diameter of reel, the number of feet divided by 1requals the diameter of the coil. As the reel is payed out the diameteris constantly being reduced while conversely during the wind up of modethe diameter is continuously increasmg. I

Motors are designed to operate between full field. and weakest field toprovide a range from base speed to the highest speed. Depending upon themanufacturer's specification, operation beyond weakest field may bepermitted for certain voltages ranges and for limited periods of time.

The motor field controller (FIG. 3) operates such that above the reeldiameter Do corresponding to the weakest field flux:

Below the reel diameter Do at weakest field:

constant The potentiometer 64 is adjusted to a voltage e equal to thevoltage signal D when the reel diameter is at machine during pay-off,and a motoring machine during rewind. At weak field, the motor 18 willrun at higher speeds and voltage to accommodate the line speed and'thiscould possible cause electrical and meweakest field. The input signal tothe proportional integral controller 60 is the voltage at node A dividedby the ohmic magnitude of resistor 58. At large reel diameters thevoltage signal D is greater than e, and the flux d is directlyproportional to the diameter D. As the reel diameter decreases, theinput signal decreases proportionally. When the reel diameter reaches Doand then decreases below the weakest field diameter Do, the voltage edetermines the input signal to proportional integral controller 60. Thisinput signal is the voltage e, divided by the ohmic magnitude ofresistor 58. Thus the input signal is a constant at and below theweakest field.

Because of motor field flux saturation, the motor field flux is notdirectly proportional to motor field current. A function generator isrequired to convert the motor field feedback signal to a voltageproportional to the motor field flux. The function generator 84 modifiesthe motor field feedback signal from the current sensor; 82, producingthe output voltage vs input voltage curve shown in FIG. 4. The outputvoltage is then proportional to the motor field flux. The functiongenerator 84 shapes the feedback signal to compensate for ironsaturation of the motor field. The shape of the curve depends on thedesign characteristics of the motor. The curve of FIG. 4 is derived fromthe motor field flux saturation curve which is peculiar to theparticular motor being utilized. 1

The output of the proportional integral which is to provide gatingpulses to fire the thyristors of the thyristor power module 74 whichconverts the ac line voltage to a controlled d.c. output voltage. Thediameter signal D generated from the circuitry of FIG. 2 is also appliedto the armature current controller of FIG. 5. 1

Briefly, to review, the requirements for armature current (for constanttension at a given line speed) are: (a) above weakest field reeldiameterDo, the current is constant, and (b) at and below weakest fieldreel diameter Do, the armature current is proportional tothediameter D.

When the diameter signal D is greater than weakest field diameter Do,The voltage at node B is equal to e (neglecting the dropacross thediode), and the armature current is constant depending upon theoperators setting for potentiometer 98. As the reel diameter decreases,the voltage D becomes less than voltage e, and node B matches thevoltage signal D. The input signal to the proportional integralcontroller 100 therefrom decreases and armature current decreases. Thusthe ar-' mature current reference decreases proportionally with the reelcoil diameter.

The armature current controller 100 also receives an controller 60 isapplied to the gating amplifier 72 the function of chanical damage tothe motor. Depending upon the -manufacturers rating, some overloadvoltage may be tolerated brief periods. Howevena's indicated earlier aprimary objective of this invention'is to enable a lower h.p. motor tobe utilized, so most likely the smaller motor will have limitedovervoltage range. Thus it is likely that the line speed will have to bereduced. The operator can determine when the line speed should bereduced and by how much. At the operator's console 22 (FIG. 1) thesignal D0 tells him the line speed should be reduced. The speed of theline speed pace motor is then reduced while the operator monitors thereel motor speed by means of the signal from the tachometer generator20. At smaller reel diameters the reduction in line speed is relativelyOsmall so that this is an acceptable solution for most process lineapplications.

Completing the details for the circuitry of FIG. 5, the

potentiometer 92 is set to a voltage e which is equal to the voltagesignal Do when weakest field is reached. Resistor 96 and thepotentiometer 98 are selected so that the voltage at node B (with theleads to the cathodes of diodes 90,94 open) is greater than voltage eResistor 102 is selected so that with the voltage at node B equal tovoltage e by turning the potentiometer full clockwise 100 percent ratedarmature current will obtain. The potentiometer 92 and diode 94' areincluded as part of the circuitry to' facilitate set up and fieldadjustments.

I claim:

1. A constant tension control system for a direct current reel drivemotor operating to wind or unwind a coil of strip material moving at apreset line speed comprismg:

a. means for providing a signal which is a function of the instantaneouscoil diameter D, the coil having a finite diameter D0 when the reelmotor is at the weakest field flux;

b. means coupled to receive said diameter signal D and to deliver afield control outputsignal which is coupled to the field of said reelmotor, so that the field flux 1b is a function of the instantaneousdiameter D when D Do, and the field flux 4) is constant when D D0;

c. means coupled to receive said diameter signal D and to deliver anarmature current output control signal which is coupled to the armatureof the reel motor so that when D Do the armature current la is constant,and lais a function of D when D d. means coupled to receive saiddiameter signal Do toprovide an output signal to enable the preset linespeed to be reduced when D 5 D0.

2. A constant tension control systemaccording to claim 1 wherein a. saiddiameter signal means comprises divider means having dividend anddivisor inputs, and a 3,749,988 r v i i quotient output, said inputsbeing connected to receive a line speed signal and a reel motorrotational speed signal respectively, the quotient output being afunction of the instantaneous diameter D.

3. A constant tension control system according to claim 1 wherein b.said motor field control means comprises mixer means and an operationalamplifier operated as a proportional integrator, said mixer means havingdual inputs and an output, said inputs receiving 10 1 said diametersignal D and a constant magnitude signal which is a function of D0, saidmixer means output being a function of D when D Do and being equal tosaid constant magnitude signal for all magnitudes of D Do, saidoperational amplifier having an input for receiving said mixer output,and an output for delivering said field control output signal.

4. A constant tension control system according to amplifier having aninput for receiving said mixer output and an output for delivering saidarmature current control output signal.

5. A constant tension control system according to claim 2 comprising i Ie. means for memorizing the instantaneous diameter D, comprising anoperational amplifien'a motoroperated potentiometer comprises a motorand a plurality of potentiometers', the potentiometers have respectivewiper contacts which are ganged together and coupled to the motor shaftso as to be displaced in unison, the operational amplifier beingoperated as an integrator, and having an input and an output, the inputbeing connected to receive said quotient output, the output beingconnected to claim 1 wherein 2O energize said potentiometer motor, oneof said wipc. said motor armature current control means combeingconnected in a feedback P to said P- erational amplifier input, theinstantaneous positions of said wipers, being a voltage magnitude whichis a direct function of the instantaneous diameter D.

I t t I

1. A constant tension control system for a direct current reel drivemotor operating to wind or unwind a coil of strip material moving at apreset line speed comprising: a. means for providing a signal which is afunction of the instantaneous coil diameter D, the coil having a finitediameter Do when the reel motor is at the weakest field flux; b. meanscoupled to receive said diameter signal D and to deliver a field controloutput signal which is coupled to the field of said reel motor, so thatthe field flux phi is a function of the instantaneous diameter D whenD > Do, and the field flux phi is constant when D < OR = Do; c. meanscoupled to receive said diameter signal D and to deliver an armaturecurrent output control signal which is coupled to the armature of thereel motor so that when D > Do the armature current Ia is constant, andIa is a function of D when D < OR = Do; and d. means coupled to receivesaid diameter signal Do to provide an output signal to enable the presetline speed to be reduced when D < OR = Do.
 2. A constant tension controlsystem according to claim 1 wherein a. '' said diameter signal meanscomprises divider means having dividend and divisor inputs, and aquotient output, said inputs being connected to receive a line speedsignal and a reel motor rotational speed signal respectively, thequotient output being a function of the instantaneous diameter D.
 3. Aconstant tension control system according to claim 1 wherein b. '' saidmotor field control means comprises mixer means and an operationalamplifier operated as a proportional integrator, said mixer means havingdual inputs and an output, said inputs receiving said diameter signal Dand a constant magnitude signal which is a function of Do, said mixermeans output being a function of D when D > Do and being equal to saidconstant magnitude signal for all magnitudes of D < or = Do, saidoperational amplifier having an input for receiving said mixer output,and an output for delivering said field control output signal.
 4. Aconstant tension control system according to claim 1 wherein c. '' saidmotor armature current control means comprises mixer means and anoperational amplifier operated as a proportional integrator, said mixermeans having dual inputs and an output, said outputs receiving saiddiameter signal D, and a constant magnitude signal which is a functionof Do, said mixer means output being equal to said constant signal whenD > Do, and being a function of D for all magnitudes of D < or = Do,said operational amplifier having an input for receiving said mixeroutput and an output for delivering said armature current control outputsignal.
 5. A constant tension control system according to claim 2comprising e. means for memorizing the instantaneous diameter D,comprising an operational amplifier, a motor-operated potentiometercomprises a motor and a plurality of potentiometers, the potentiometershave respective wiper contacts which are ganged together and coupled tothe motor shaft so as to be displaced in unison, the operationalamplifier being operated as an integrator, and having an input and anoutput, the input being connected to receive said quotient output, theoutput being connected to energize said potentiometer motor, one of saidwipers being connected in a feedback path to said operational amplifierinput, the instantaneous positions of said wipers, being a voltagemagnitude which is a direct function of the instantaneous diameter D.